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Answer :
Sure! Let's break down the expression step-by-step:
We need to find the value of [tex]\(\left((-625)^2\right)^{-\frac{1}{8}}\)[/tex].
Step 1: Calculate [tex]\((-625)^2\)[/tex]
When we square [tex]\(-625\)[/tex], we get:
[tex]\[
(-625)^2 = 625 \times 625 = 390625
\][/tex]
Step 2: Calculate [tex]\(390625^{-\frac{1}{8}}\)[/tex]
The expression [tex]\(a^{-b}\)[/tex] means [tex]\( \frac{1}{a^b} \)[/tex]. So, [tex]\(390625^{-\frac{1}{8}}\)[/tex] is equivalent to [tex]\(\frac{1}{390625^{\frac{1}{8}}}\)[/tex].
Finding the eighth root of 390625 gives us 5 because:
[tex]\[
5^8 = 390625
\][/tex]
Therefore, [tex]\(\frac{1}{390625^{\frac{1}{8}}}\)[/tex] simplifies to [tex]\(\frac{1}{5}\)[/tex].
So the expression [tex]\(\left((-625)^2\right)^{-\frac{1}{8}}\)[/tex] evaluates to [tex]\(\frac{1}{5}\)[/tex].
Thus, the correct answer is:
D) [tex]\(\frac{1}{5}\)[/tex]
We need to find the value of [tex]\(\left((-625)^2\right)^{-\frac{1}{8}}\)[/tex].
Step 1: Calculate [tex]\((-625)^2\)[/tex]
When we square [tex]\(-625\)[/tex], we get:
[tex]\[
(-625)^2 = 625 \times 625 = 390625
\][/tex]
Step 2: Calculate [tex]\(390625^{-\frac{1}{8}}\)[/tex]
The expression [tex]\(a^{-b}\)[/tex] means [tex]\( \frac{1}{a^b} \)[/tex]. So, [tex]\(390625^{-\frac{1}{8}}\)[/tex] is equivalent to [tex]\(\frac{1}{390625^{\frac{1}{8}}}\)[/tex].
Finding the eighth root of 390625 gives us 5 because:
[tex]\[
5^8 = 390625
\][/tex]
Therefore, [tex]\(\frac{1}{390625^{\frac{1}{8}}}\)[/tex] simplifies to [tex]\(\frac{1}{5}\)[/tex].
So the expression [tex]\(\left((-625)^2\right)^{-\frac{1}{8}}\)[/tex] evaluates to [tex]\(\frac{1}{5}\)[/tex].
Thus, the correct answer is:
D) [tex]\(\frac{1}{5}\)[/tex]
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